Multi-Period Prediction of Solar Radiation Using ARMA and ARIMA Models

Multi-Period Prediction of Solar Radiation Using

ARMA and ARIMA Models

Ilhami Colak1, Mehmet Yesilbudak2, Naci Genc3, Ramazan Bayindir4

1

Istanbul Gelisim University, Faculty of Engineering and Architecture, Department of Mechatronic Engineering, 34315, Istanbul, Turkey. 2_{Nevsehir Haci Bektas Veli University, Faculty of Engineering and Architecture, Department of Electrical and Electronics Engineering, 50300,}

Nevsehir, Turkey. 3

Yuzuncu Yil University, Faculty of Engineering and Architecture, Department of Electrical and Electronics Engineering, 65080, Van, Turkey. 4

Gazi University, Faculty of Technology, Department of Electrical and Electronics Engineering, 06500, Ankara, Turkey. icolak@gelisim.edu.tr, myesilbudak@nevsehir.edu.tr, nacigenc@yyu.edu.tr, bayindir@gazi.edu.tr

Abstract—Due to the variations in weather conditions, solar power integration to the electricity grid at a high penetration rate can cause a threat for the grid stability. Therefore, it is required to predict the solar radiation parameter in order to ensure the quality and the security of the grid. In this study, initially, a 1-h time series model belong to the solar radiation parameter is created for multi-period predictions. Afterwards, autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models are compared in terms of the goodness-of-fit value produced by the log-likelihood function. As a result of determining the best statistical models in multi-period predictions, one-period, two-period and three-period ahead predictions are carried out for the solar radiation parameter in a comprehensive way. Many feasible comparisons have been made for the solar radiation prediction.

Keywords—Solar radiation; multi-period prediction; ARMA; ARIMA

I. INTRODUCTION

According to the Renewables Global Status Report 2015 [1], the solar photovoltaic capacity in the world has reached to 177 GW in 2014. Besides, it is projected to be 1721 GW by 2030 [2]. So, solar energy is one of the most promising renewable energy sources and the solar power integration in electricity grids is constantly increasing all over the world [3, 4]. However, solar power shows high variations due to the intermittency nature of solar energy [5]. For this reason, the generation schedules are mostly planned for hourly, daily, weekly, etc. in order to ensure reliable operation and economic dispatch of solar power systems [6]. As a result, it is indispensible to predict the solar radiation parameter in solar power plants. Since, the solar radiation parameter is considered as the most significant indicator for the solar energy conversion and for the sizing of stand-alone solar systems [7, 8]. The world map of global horizontal irradiation is shown in Figure 1.

Several authors have proposed various models for solar radiation prediction in the literature. Yadav et al. and Qazi et

al. reviewed artificial neural network techniques for solar

radiation prediction and the prediction accuracy was found to be dependent on input parameter combinations, training algorithms and architecture configurations [9, 10]. Chen et al. investigated the feasibility of support vector machines in order

to predict solar radiation using air temperatures and the impacts of inputs and kernel functions on the prediction accuracy were determined [11]. Bhardwaj et al. analyzed the inter-dependence of solar radiation and other meteorological parameters by the generalized fuzzy model and short-term prediction assessments were made under different climatic conditions [12]. Salcedo-Sanz et al. evaluated the effectiveness of temporal Gaussian process regression for the estimation of daily global solar radiation and a time-based composite covariance was presented in order to account for the relevant seasonal variations [13]. Licciardi et al. used the nonlinear principal component analysis to reduce the dimensionality of spatiotemporal input vector and improved the forecasting of ground horizontal irradiance from satellite-based images [4]. Wu et al. proposed a multi-model framework for short-term prediction of solar radiation time series and the nonlinear relationship of different patterns were modeled for capturing the general trend of whole series [14].

Fatemi et al. utilized the zenith angle along with exponentially

weighted recursive least squares method and the seasonal/daily effects in solar radiation data were removed [15]. Huang et. al. detected the solar irradiance fluctuations from cloud movements and the solar power volatility was mitigated in electric grids [16]. In addition, numerous hybrid methods such as grey-based support vector regression [17], wavelet-based recurrent neural networks [7], wavelet-based support vector machines [18], Kalman-based radial basis functions [19], Kalman-based neuro-fuzzy inference systems [20] etc. were employed for solar radiation prediction in the literature. Apart from these hybrid methods, many numerical weather prediction models such as MM5 (Mesoscale Modeling) [21], GFS (Global Forecasting System) [22], WRF (Weather Research and Forecasting) [23], NAM (North American Mesoscale) [24], ECMWF (European Centre for Medium-Range Weather Forecasts) [25], etc. were also utilized for solar radiation prediction in the literature.

The main objective of this study is to analyze multi-period predictions of solar radiation using ARMA and ARIMA models in detail. As a distinct contribution to the literature, not only mean absolute errors and mean absolute percentage errors but also improvement percentages of prediction results are revealed in multi-period predictions. Since, most of studies in the literature ignore the persistence comparison that is 2015 IEEE 14th International Conference on Machine Learning and Applications

2015 IEEE 14th International Conference on Machine Learning and Applications 2015 IEEE 14th International Conference on Machine Learning and Applications

employed as a reference analysis for conducting a proper benchmark test. In addition, many reasonable comparisons have been realized among ARMA and ARIMA models in terms of their prediction accuracy.

Fig. 1. The world map of global horizontal irradiation [26] II. MULTI-PERIOD PREDICTION APPROACH

In this study, autoregressive moving average and autoregressive integrated moving average models have been used for multi-period predictions. ARMA and ARIMA models are applied to stationary and non-stationary stochastic time series data, respectively and ARIMA process is an ARMA process for the differenced time series [27]. The detailed theoretical instructions about ARMA and ARIMA models are included in [28]. The Log-Likelihood Function (LLF) has been employed for the goodness-of-fit determination among ARMA and ARIMA models in this paper. More statistical formulations about the LLF are found in [29]. In the log-likelihood criterion, the closer the log-log-likelihood value is to zero, the more likely it is that the parameters could produce the observed data [30]. The LLF values belong to ARMA and ARIMA models are given in Table I. It is obvious that ARMA(1,2) and ARIMA(2,2,2) models achieved the optimum log-likelihood values in here.

TABLE I. LLF VALUES OF ARMA AND ARIMA MODELS

Statistical Model LLF Value Statistical Model LLF Value

ARMA(1,1) -97.95 ARIMA(1,2,1) -68.63 ARMA(1,2) -96.82 ARIMA(2,1,1) -82.00 ARMA(2,1) -98.84 ARIMA(1,2,2) -68.14 ARMA(2,2) -97.77 ARIMA(2,1,2) -81.40 ARIMA(1,1,1) -81.94 ARIMA(2,2,1) -68.12 ARIMA(1,1,2) -81.31 ARIMA(2,2,2) -68.08 In addition to the mentioned specifications above, mean absolute error (MAE) and mean absolute percentage error (MAPE) have been utilized for measuring the prediction accuracy of ARMA(1,2) and ARIMA(2,2,2) models in this study. These error metrics are given in Equation (1) and (2) by assuming ݊ as the number of test data, ݕ௜ as the observed solar radiation and ý௜ as the predicted solar radiation [31]. Besides, ARMA(1,2) and ARIMA(2,2,2) models have been compared with the persistence model in terms of the improvement percentage of prediction results. Since, the persistence comparison is made as a reference analysis for proper benchmark tests in the literature [32]. As well, the last three observations have been used as the most important predictors in each multi-period prediction approach. It should be noted

that other ARMA and ARIMA models that are not included in Table I are not employed owing to their high errors.

 ൌͳ ෍ȁý୧െ ›୧ȁ ୬ ୧ୀଵ (1)  ൌͳ_෍ ൬ฬý୧െ ›_› ୧ ୧ ฬ൰ ൈ ͳͲͲ ୬ ୧ୀଵ (2)

In multi-period predictions, a 1-h time series model is created for the solar radiation parameter. Figure 2 shows the concept of a multi-period prediction with the 1-h time series model [33, 34]. In Figure 2(a), the mean hourly solar radiation at the subsequent interval of [03:00, 04:00) is predicted using the mean observed hourly solar radiation at the intervals of [00:00, 01:00), [01:00, 02:00) and [02:00, 03:00). In Figure 2(b), the mean hourly solar radiation at the subsequent interval of [04:00, 05:00) is predicted using the mean observed hourly solar radiation at the intervals of [01:00, 02:00), [02:00, 03:00) and the previously predicted solar radiation at the interval of [03:00, 04:00). In Figure 2(c), the mean hourly solar radiation at the subsequent interval of [05:00, 06:00) is predicted using the mean observed hourly solar radiation at the interval of [02:00, 03:00) and the previously predicted solar radiation at the intervals of [03:00, 04:00), [04:00, 05:00).

(a) One-period ahead prediction

(b) Two-period ahead prediction

(c) Three-period ahead prediction

Fig. 2. The concept of a multi-period prediction with the 1-h time series model

III. MULTI-PERIOD PREDICTION RESULTS

In this section, it should be noted that one-, two- and three-period ahead predictions represent the 1-h, 2-h and 3-h ahead solar radiation predictions, respectively. In addition, the starting time of the prediction process is different from each other in the conducted prediction analyses. Since, the solar radiation values averaged at the intervals of [00:00, 00:01), [01:00, 02:00) and [02:00, 03:00] are equal to zero. So, the prediction process starts at the time that has the first positive value measured for the solar radiation parameter and it continues up to the time that has the first zero value measured for the solar radiation parameter. As mentioned in the previous section, ARMA(1,2) and ARIMA(2,2,2) models are only considered in multi-period predictions for the reason of achieving optimum log-likelihood values.

A. One-Period Ahead Prediction Results

One-period ahead prediction results and errors for solar radiation parameter are illustrated in Figures 3 and 4, respectively. The MAPEs of ARMA(1,2) and ARIMA(2,2,2) models are found as 18.11% and 7.87%, respectively. However, it is obtained as 38.44% for the persistence model. In case of making a benchmark test with respect to the persistence model, ARMA(1,2) and ARIMA(2,2,2) models have improved the prediction results at the rates of 52.89% and 79.53%, respectively. From another perspective, it is obvious that ARIMA(2,2,2) model has outperformed ARMA(1,2) model in terms of the one-period ahead prediction of solar radiation parameter. In addition, the mean absolute error of ARIMA(2,2,2) model is acquired as 37.95 W/m2. The absolute percentage errors of ARMA(1,2) and ARIMA(2,2,2) models in one-period ahead prediction are listed in Table II.

Fig. 3. One-period ahead prediction results for solar radiation

Fig. 4. One-period ahead prediction errors for solar radiation

TABLE II. ABSOLUTE PERCENTAGE ERRORS OF ARMA(1,2) AND ARIMA(2,2,2) MODELS IN ONE-PERIOD AHEAD PREDICTION

Time ARMA(1,2) ARIMA(2,2,2) Time ARMA(1,2) ARIMA(2,2,2)

04:00 57.65 47.71 11:00 0.10 3.16 05:00 14.45 7.08 12:00 12.93 9.93 06:00 14.60 2.29 13:00 3.15 9.34 07:00 6.78 3.40 14:00 16.48 4.77 08:00 4.89 1.36 15:00 26.58 1.39 09:00 0.52 3.45 16:00 74.56 5.11 10:00 2.73 3.34 17:00 - -

B. Two-Period Ahead Prediction Results

Two-period ahead prediction results and errors for solar radiation parameter are depicted in Figures 5 and 6, respectively. ARMA(1,2) and ARIMA(2,2,2) models produce the MAPEs of 43.24% and 16.06%, respectively. Nonetheless, the persistence model gives it as 67.37%. In the stage of conducting a benchmark test against the persistence model, the improvement percentages for the prediction results have been achieved as 35.82% and 76.16% for ARMA(1,2) and ARIMA(2,2,2) models, respectively. Apart from these inferences, it is clear that ARIMA(2,2,2) model has surpassed ARMA(1,2) model in the two-period ahead prediction of solar radiation parameter. As well, ARIMA(2,2,2) model brings out the mean absolute error of 88.51 W/m2. The absolute percentage errors of ARMA(1,2) and ARIMA(2,2,2) models in two-period ahead prediction are listed in Table III.

Fig. 5. Two-period ahead prediction results for solar radiation

Fig. 6. Two-period ahead prediction errors for solar radiation

TABLE III. ABSOLUTE PERCENTAGE ERRORS OF ARMA(1,2) AND ARIMA(2,2,2) MODELS IN TWO-PERIOD AHEAD PREDICTION

Time ARMA(1,2) ARIMA(2,2,2) Time ARMA(1,2) ARIMA(2,2,2)

04:00 - - 11:00 5.16 5.52 05:00 71.88 56.99 12:00 12.71 0.69 06:00 32.57 9.58 13:00 31.84 20.36 07:00 28.02 1.09 14:00 24.17 26.00 08:00 15.71 0.54 15:00 75.23 48.80 09:00 9.02 6.63 16:00 210.86 4.48 10:00 1.75 12.11 17:00 - -

C. Three-Period Ahead Prediction Results

Three-period ahead prediction results and errors for solar radiation parameter are picturized in Figures 7 and 8, respectively. Similar to the one-period and two-period ahead prediction results, ARIMA(2,2,2) model provides lower prediction errors than ARMA(1,2) and persistence models. MAPEs of them are determined as 32.07%, 71.67% and 92.81%, respectively. As a result of comparing ARMA(1,2) and ARIMA(2,2,2) models with the persistence model on the basis of the enhancement occurred in the prediction results, the improvement ratios have been accomplished as 22.78% and 65.45%, respectively. Besides, ARIMA(2,2,2) model have gave the mean absolute error of 172.07 W/m2. The absolute percentage errors of ARMA(1,2) and ARIMA(2,2,2) models in three-period ahead prediction are listed in Table IV.

Fig. 7. Three-period ahead prediction results for solar radiation

Fig. 8. Three-period ahead prediction errors for solar radiation

TABLE IV. ABSOLUTE PERCENTAGE ERRORS OF ARMA(1,2) AND ARIMA(2,2,2) MODELS IN THREE-PERIOD AHEAD PREDICTION

Time ARMA(1,2) ARIMA(2,2,2) Time ARMA(1,2) ARIMA(2,2,2)

04:00 - - 11:00 3.91 20.95 05:00 - - 12:00 20.31 17.88 06:00 80.17 63.26 13:00 31.52 1.62 07:00 45.44 14.69 14:00 70.90 40.34 08:00 38.29 2.12 15:00 90.34 62.65 09:00 21.55 18.26 16:00 377.05 93.37 10:00 8.92 17.58 17:00 - - IV. CONCLUSIONS

In this study, ARMA and ARIMA models are employed for one-period, two-period and three-period ahead predictions of the solar radiation parameter. In case of comparing all of multi-period prediction results, ARIMA(2,2,2) model provides the lowest mean absolute percentage errors and leads to the largest improvement percentages. It is followed by ARMA(1,2) and persistence models in terms of the prediction performance, respectively. In other words, the persistence model shows the worst prediction accuracy in all of multi-period predictions. From a different perspective, it is revealed for persistence, ARMA(1,2) and ARIMA(2,2,2) models that the mean absolute percentage error increases and the improvement percentage decreases as the prediction period progresses. In addition, the first prediction errors are usually high in all of multi-period predictions for the reason that two of the first three important predictors equal to zero at the beginning of the prediction process.

In future studies, not only the solar radiation parameter but also air temperature and sunshine duration parameters should be considered for multi-time series and multi-time scale modeling.

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